Aircraft: B757
Phase of flight: Descent; Cruise; Taxi
Anomaly: Deviation - Altitude Excursion From Assigned Altitude; Deviation - Speed All Types; Deviation / Discrepancy - Procedural Published Material / Policy; Deviation / Discrepancy - Procedural Clearance

I was the PIC and pilot flying Aircraft X from ZZZ to ZZZ1; and made numerous operational errors due to schedule pressures and not having had adequate sleep. Although; I'd had a large cup of coffee while waiting for the aircraft to be brought over to us after it was unloaded; I felt fine when signing the fitness for duty statement. The sleep issue was due to being 14 time zones from home and not acclimating like I used to when I was younger. I had two days in ZZZ before we flew but did not sleep more than 3 to 3.5 hours at a time; and took a few naps as well. The first error occurred prior to departure when I noticed the POAT (Planned Outside Air Temperature) of 28 degrees Celsius was below the ATIS OAT (Outside Air Temperature) of 32 degrees Celsius. I called Dispatch and requested an updated PMRTW (Planned Maximum Runway Takeoff Weight) based on using a POAT of 35 degrees Celsius; in compliance with [company policy]. The dispatcher did not seem to know what I was referring to; so I walked him through the process and recorded the updated information on my TLR (Takeoff and Landing Report); but failed to request an amendment to my release for the new lower PMRTW; thinking I would do that when I got the final release.The second error occurred at scheduled departure time when I was handed the logbook with three new ACIs (Administrative Control Items) in it. I called Dispatch and amended the release to reflect the addition of the new ACIs; and revised the Weight and Balance Load Manifest to reflect a total weight of 884 kg. versus 870 kg. as shown in the release. We were now past our scheduled departure time so I requested my release and a weather update via ACARS; but forgot to amend the release for the revised PMRTW. We completed our checklists; pushed back and started engines. We taxied out and departed without receiving our requested weather update from Dispatch; and I did not catch it until enroute.The third error occurred about 30 to 50 minutes into the flight when I was looking at the enroute wind data in cruise and made the mistake of entering a couple waypoints of winds while the PM (Pilot Monitoring) was heads down entering the departure report in ACARS. My error; I should not have been heads down at the same time as him. For some unexplained reason; the FMCS (Flight Management Computer System) sequenced past our active way point and the aircraft began a turn away from the course. The PM and I both noticed this at roughly the same time; and instead of merely selecting HDG SEL; and turning the aircraft back to the desired course; I disconnected the autopilot and manually turned the aircraft. In so doing; I began a gentle climb which I did not notice until I heard the aural altitude alert. I don't recall exactly how far off our altitude I was; but immediately began a descent back to 27;000 feet. I requested the PM to re-enter the correct active waypoint and reset the automation; which he did. This all occurred in less than 30 seconds; and nothing was mentioned to us by ATC. We were in [location] with no traffic in sight; or displayed on TCAS.The fourth error occurred on descent into ZZZ1 approximately 15 minutes before landing. ZZZ1 is not in the B757 FMCS navigation database in the airplane; and is surrounded by terrain. [Center] initially gave us direct to the ZZZZZ1 after ZZZZZ; so I created a fix 30 miles prior to ZZZZZ1 for descent planning purposes. After beginning our descent to 12;000 feet; we were handed off to ZZZ1 Approach; who attempted to give us three different fixes; ZZZZZ2; ZZZZZ3; and ZZZZZ4 for the approach to Runway XX. The ZZZZZ2 in our navigation database was the wrong ZZZZZ2; and was located 8;495 miles away. ZZZZZ3 wasn't in the database; but fortunately ZZZZZ4 was. We proceeded towards ZZZZZ4 and created another descent planning fix prior to ZZZZZ4 and were cleared to descend to 9;000 ft. The transition level was 11;000 ft. and as we passed 11;000 ft.; we reset the altimeters and completed the Descent and Approach checklist. At this time we were cleared to descend to 6;000 ft.; and as PF (Pilot Flying) I reset the altitude selector on the MCP (Mode Control Panel). As we descended through 10;000 ft.; I realized I hadn't started decelerating to 250 KIAS and erroneously selected the HDG HOLD button in lieu of the ALT HOLD button on the MCP in my attempt to arrest the descent. It took a few seconds to register in my foggy brain what I had done; and reselected ALT HOLD and LNAV. We decelerated to 245 KIAS and then continued the descent to 6;000 ft.The fifth error occurred after parking on shutdown. After setting the parking brake; I verified we had the APU running; shut down the engines; turned off the red anti-collision light; and as I turned off the SEATBELT sign; called for the Shutdown checklist. The PM apparently did not hear me; and then I noticed some apparent confusion among the ground crew. I gave them the chocks in hand signal and they proceeded to chock the aircraft. By this time I had let my guard down; failed to realize that we had not completed the Shutdown checklist and proceeded to complete our post-flight duties. The aircraft wasn't going to depart again for a few hours so the maintenance technicians fueled the aircraft; shut it down and then teased me about leaving the hydraulic pumps and fuel pumps on (they did turn them OFF when they depowered the aircraft). I was tired and embarrassed by my errors; having never screwed up like this before. Nothing was said by either ATC unit regarding the altitude and speed deviations; but I believed it best to self-disclose them rather than try to hide from them and hope nothing comes of them.Obviously; I need to slow down and ensure I check all of the boxes. I failed to do so on this flight; and will try to ensure I do so in the future. One thing which would be extremely helpful would be for Dispatch to automatically send us the weather update with our Release via ACARS without being asked. However; we as pilots should also be asking for that without a reminder. It would also be highly beneficial to have airports like ZZZ1 (into which we have operated numerous times) in the aircraft navigation database. Additionally; I obviously have a greater understanding now why there is a delineation between PM and PF duties with respect to the FMC. Finally; I'm not sure how to best address the fatigue issue (other than sleeping more). Obviously I need more sleep than I've been getting; and am open to guidance on being able to more accurately assess my fitness for duty. B757 Captain reported errors made in preflight; cruise; descent and shutdown phases of flight due to fatigue and incomplete navigation database.